## Paper: An n! Lower Bound on Formula Size (at LICS 2001)

**Micah Adler Neil Immerman**

### Abstract

We introduce a new Ehrenfeucht-Fraïssé game for proving lower bounds on the size of first-order formulas. Up until now such games have only been used to prove bounds on the operator depth of formulas, not their size. We use this game to prove that the CTL+ formula {\rm Occur}_n\equiv{\bf E}[{\bf F}p_1\wede{\bf F}_2\wedge\cdots\wedge{\bf F}p_n] which says that there is a path along which the predicates p1 through pn occur in some order, requires size n! to express in CTL. Our lower bound is optimal. It follows that the succinctness of CTL+ with respect to CTL is exactly T(n)!. Wilke had shown that the succinctness was at least exponential [Wil99]. We also use our games to prove an optimal O(n) lower bound on the number of boolean variables needed for a weak reachability logic ({\cal RL}^w) to polynomially embed the language LTL. The number of booleans needed for full reachability logic RL and the transitive closure logic FO2(TC) remain open [IV97, AI00].

### BibTeX

@InProceedings{AdlerImmerman-AnnLowerBoundonForm, author = {Micah Adler and Neil Immerman}, title = {An n! Lower Bound on Formula Size}, booktitle = {Proceedings of the Sixteenth Annual IEEE Symposium on Logic in Computer Science (LICS 2001)}, year = {2001}, month = {June}, pages = {197--206}, location = {Boston, MA, USA}, publisher = {IEEE Computer Society Press} }